Chapter 11 Angular Momentum; General Rotation

Angular Momentum—Objects Rotating About a Fixed Axis Vector Cross Product; Torque as a Vector Angular Momentum of a Particle Angular Momentum and Torque for a System of Particles; General Motion Angular Momentum and Torque for a Rigid Object Units of Chapter 11

Conservation of Angular Momentum The Spinning Top and Gyroscope Rotating Frames of Reference; Inertial Forces The Coriolis Effect Units of Chapter 11

11-1 Angular Momentum—Objects Rotating About a Fixed Axis The rotational analog of linear momentum is angular momentum, L: Then the rotational analog of Newton’s second law is: This form of Newton’s second law is valid even if I is not constant.

11-1 Angular Momentum—Objects Rotating About a Fixed Axis In the absence of an external torque, angular momentum is conserved: More formally, the total angular momentum of a rotating object remains constant if the net external torque acting on it is zero.

11-1 Angular Momentum—Objects Rotating About a Fixed Axis This means: Therefore, if an object’s moment of inertia changes, its angular speed changes as well.

11-1 Angular Momentum—Objects Rotating About a Fixed Axis Example 11-1: Object rotating on a string of changing length. A small mass m attached to the end of a string revolves in a circle on a frictionless tabletop. The other end of the string passes through a hole in the table. Initially, the mass revolves with a speed v 1 = 2.4 m/s in a circle of radius R 1 = 0.80 m. The string is then pulled slowly through the hole so that the radius is reduced to R 2 = 0.48 m. What is the speed, v 2, of the mass now?

11-1 Angular Momentum—Objects Rotating About a Fixed Axis Example 11-2: Clutch. A simple clutch consists of two cylindrical plates that can be pressed together to connect two sections of an axle, as needed, in a piece of machinery. The two plates have masses M A = 6.0 kg and M B = 9.0 kg, with equal radii R 0 = 0.60 m. They are initially separated. Plate M A is accelerated from rest to an angular velocity ω 1 = 7.2 rad/s in time Δt = 2.0 s. Calculate (a) the angular momentum of M A, and (b) the torque required to have accelerated M A from rest to ω 1. (c) Next, plate M B, initially at rest but free to rotate without friction, is placed in firm contact with freely rotating plate M A, and the two plates both rotate at a constant angular velocity ω 2, which is considerably less than ω 1. Why does this happen, and what is ω 2 ?

11-1 Angular Momentum—Objects Rotating About a Fixed Axis Example 11-3: Neutron star. Astronomers detect stars that are rotating extremely rapidly, known as neutron stars. A neutron star is believed to form from the inner core of a larger star that collapsed, under its own gravitation, to a star of very small radius and very high density. Before collapse, suppose the core of such a star is the size of our Sun (r ≈ 7 x 10 5 km) with mass 2.0 times as great as the Sun, and is rotating at a frequency of 1.0 revolution every 100 days. If it were to undergo gravitational collapse to a neutron star of radius 10 km, what would its rotation frequency be? Assume the star is a uniform sphere at all times, and loses no mass.

11-1 Angular Momentum—Objects Rotating About a Fixed Axis Angular momentum is a vector; for a symmetrical object rotating about a symmetry axis it is in the same direction as the angular velocity vector.

11-1 Angular Momentum—Objects Rotating About a Fixed Axis Example 11-4: Running on a circular platform. Suppose a 60-kg person stands at the edge of a 6.0- m-diameter circular platform, which is mounted on frictionless bearings and has a moment of inertia of 1800 kg·m 2. The platform is at rest initially, but when the person begins running at a speed of 4.2 m/s (with respect to the Earth) around its edge, the platform begins to rotate in the opposite direction. Calculate the angular velocity of the platform.

11-1 Angular Momentum—Objects Rotating About a Fixed Axis Conceptual Example 11-5: Spinning bicycle wheel. Your physics teacher is holding a spinning bicycle wheel while he stands on a stationary frictionless turntable. What will happen if the teacher suddenly flips the bicycle wheel over so that it is spinning in the opposite direction?

11-2 Vector Cross Product; Torque as a Vector The vector cross product is defined as: The direction of the cross product is defined by a right-hand rule:

11-2 Vector Cross Product; Torque as a Vector The cross product can also be written in determinant form:

11-2 Vector Cross Product; Torque as a Vector Some properties of the cross product:

11-2 Vector Cross Product; Torque as a Vector Torque can be defined as the vector product of the force and the vector from the point of action of the force to the axis of rotation:

11-2 Vector Cross Product; Torque as a Vector For a particle, the torque can be defined around a point O: Here, is the position vector from the particle relative to O.

Example 11-6: Torque vector. Suppose the vector is in the xz plane, and is given by = (1.2 m) m) Calculate the torque vector if = (150 N) Vector Cross Product; Torque as a Vector

11-3 Angular Momentum of a Particle The angular momentum of a particle about a specified axis is given by:

11-3 Angular Momentum of a Particle If we take the derivative of, we find: Since we have:

11-3 Angular Momentum of a Particle Conceptual Example 11-7: A particle’s angular momentum. What is the angular momentum of a particle of mass m moving with speed v in a circle of radius r in a counterclockwise direction?

11-4 Angular Momentum and Torque for a System of Particles; General Motion The angular momentum of a system of particles can change only if there is an external torque—torques due to internal forces cancel. This equation is valid in any inertial reference frame. It is also valid for the center of mass, even if it is accelerating:

11-5 Angular Momentum and Torque for a Rigid Object For a rigid object, we can show that its angular momentum when rotating around a particular axis is given by:

11-5 Angular Momentum and Torque for a Rigid Object Example 11-8: Atwood’s machine. An Atwood machine consists of two masses, m A and m B, which are connected by an inelastic cord of negligible mass that passes over a pulley. If the pulley has radius R 0 and moment of inertia I about its axle, determine the acceleration of the masses m A and m B, and compare to the situation where the moment of inertia of the pulley is ignored.

11-5 Angular Momentum and Torque for a Rigid Object Conceptual Example 11-9: Bicycle wheel. Suppose you are holding a bicycle wheel by a handle connected to its axle. The wheel is spinning rapidly so its angular momentum points horizontally as shown. Now you suddenly try to tilt the axle upward (so the CM moves vertically). You expect the wheel to go up (and it would if it weren’t rotating), but it unexpectedly swerves to the right! Explain.

11-5 Angular Momentum and Torque for a Rigid Object A system that is rotationally imbalanced will not have its angular momentum and angular velocity vectors in the same direction. A torque is required to keep an unbalanced system rotating.

11-5 Angular Momentum and Torque for a Rigid Object Example 11-10: Torque on unbalanced system. Determine the magnitude of the net torque τ net needed to keep the illustrated system turning.

11-6 Conservation of Angular Momentum If the net torque on a system is constant, The total angular momentum of a system remains constant if the net external torque acting on the system is zero.

11-6 Conservation of Angular Momentum Example 11-11: Kepler’s second law derived. Kepler’s second law states that each planet moves so that a line from the Sun to the planet sweeps out equal areas in equal times. Use conservation of angular momentum to show this.

11-6 Conservation of Angular Momentum Example 11-12: Bullet strikes cylinder edge. A bullet of mass m moving with velocity v strikes and becomes embedded at the edge of a cylinder of mass M and radius R 0. The cylinder, initially at rest, begins to rotate about its symmetry axis, which remains fixed in position. Assuming no frictional torque, what is the angular velocity of the cylinder after this collision? Is kinetic energy conserved?